1. 1 Chapter An Introduction to Problem Solving
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2. 1-3 Reasoning and Logic: An Introduction
Definitions
Conditionals and Biconditionals
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Definitions
Statement – a sentence that is either true or false, but not both.
Negation – a statement with the opposite truth value of the given statement. The negation of a true statement is a false statement. If p is true, then ~ p is false.
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Definitions
Quantifier – words such as “all”, “some”, “every”, “there exists”
Universal quantifier – applies to every element in a set. “All”, “every”, and “no” are universal quantifiers.
Existential quantifier – applies to one or more (or possibly every) element in a set. “Some” and “there exists at least one” are existential quantifiers.
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Example 1-7
Negate each of the following:
2 + 3 = 5
2 + 3 ≠ 5
A hexagon has six sides.
A hexagon does not have six sides.
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Example 1-8
Negate each of the following:
1. All students like hamburgers.
Some students do not like hamburgers.
2. Some people like mathematics.
No people like mathematics.
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Example 1-8, continued
Negate each of the following:
3. There exists a natural number x such that 3x = 6.
For all natural numbers, 3x ≠ 6.
4. For all natural numbers, 3x = 3x.
There exists a natural number x such that 3x ≠ 3x.
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Truth Tables
A symbolic system to show all possible true-false patterns for statements. This is the truth table for negation. p ~p T F
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Conjunction
A compound statement created from two given statements using the connective “and”. We use the symbol “” to represent “and”. A conjunction is true only if both statements are true; otherwise, it is false. p q
p  q T F
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Disjunction
A compound statement created from two given statements using the connective “or”. We use the symbol “” to represent “or”. A disjunction is false if both statements are false and true in all other cases. p q
p  q T F
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Example 1-9
Classify each of the following as true or false:
p: = 5 q: 2 • 3 = 6 r: = 9
p  q
p is true and q is true, so p  q is true.
2. q  r
q is true and r is false, so q  r is true.
~p  r
~p is false and r is false, so ~p  r is false.
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Example (continued)
Classify each of the following as true or false:
p: = 5 q: 2 • 3 = 6 r: = 9
4. ~p  ~q
~p is false and ~q is false, so ~p  ~q is false.
5. ~(p  q)
p  q is true, so ~(p  q) is false.
6. (p  q)  ~r
p  q is true and ~r is true, so (p  q)  ~r is true.
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13. Conditionals and Biconditionals
Conditional (or implication) – a statement expressed in the form “if p, then q.” Represented by p →q. p q
p → q T F
Hypothesis – the “if” part of the conditional
Conclusion – the “then” part of the conditional
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Any implication p → q has three related implication statements:
Statement if p, then q p → q
Converse if q, then p q → p
Inverse if not p, then not q ~p → ~ q
Contrapositive
if not q, then not p ~q → ~ p
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Example 1-11
Write the converse, inverse, and contrapositive for the following statement:
If I am in San Francisco, then I am in California.
Converse: If I am in California, then I am in San Francisco.
Inverse: If I am not in San Francisco, then I am not in California.
Contrapositive: If I am not in California, then I am not in San Francisco.
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A statement and its contrapositive are logically equivalent. p q
p → q T F ~p
~ q
~ q → ~ p F T
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Example 1-12
Use truth tables to determine if p → q  ~q → ~p. p q ~p ~q
p → q
~q → ~p T F
p → q is equivalent to ~q → ~p.
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Biconditional
The conjunction of a statement and its converse. It is written as p ↔ q and is read “p if and only if q.” p q
p → q
q → p
(p → q)  (q → p) T F
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Valid reasoning – if the conclusion follows unavoidably from true hypotheses.
Example
Hypotheses: All cats like fish.
Felix is a cat.
Conclusion: Therefore, Felix likes fish.
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We can use an Euler diagram to represent the validity of this reasoning.
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Example 1-13
Determine if the following argument is valid:
Hypotheses: In Washington, D.C., all lobbyists wear suits.
No one in Washington, D.C., over 6 ft tall wears a suit.
Conclusion: Persons over 6 ft tall are not lobbyists in Washington, D.C.
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Example (continued)
We can use an Euler diagram to represent the validity of this reasoning.
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Example (continued)
If L represents the lobbyists in Washington, D.C., and S the people who wear suits, the first hypothesis is pictured on the left.
If W represents the people in Washington, D.C., over 6 ft tall, the second hypothesis is pictured on the right.
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Example (continued)
Because people over 6 ft tall are outside the circle representing suit wearers and lobbyists are in the circle S, the conclusion is valid and no person over 6 ft tall is a lobbyist in Washington, D.C.
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25. Law of Detachment (Modus Ponens)
If the statement “if p, then q” is true, and p is true, then q is true. This is direct reasoning.
Example
Hypotheses: If it is raining, the grass is wet.
It is raining.
Conclusion: The grass is wet.
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26. Indirect Reasoning (Modus Tollens)
If a conditional is accepted as true, and the conclusion is false, then the hypothesis must be false.
Example
Hypotheses: If it is raining, the grass is wet.
The grass is not wet.
Conclusion: It is not raining.
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Chain Rule
If the statements “if p, then q” and “if q, then r” are true, then the statement “if p, then r” is true.
Example
Hypotheses: If you eat well, then you will be well.
If you are well, then you are happy.
Conclusion: If you eat well, then you are happy.
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